In mathematics , a Hodge structure is an algebraic structure that generalizes the Hodge decomposition of the cohomology of compact Kähler manifolds. Hodge structures have a variety of uses in complex and algebraic geometry.
Definitions
A Hodge decomposition of a real vector space is a decomposition
with for everyone .
A Hodge structure is a real vector space together with a Hodge decomposition.
A pure Hodge structure by weight is a Hodge structure with
In general, one has a weight breakdown for a Hodge structure
With
A whole Hodge structure (or rational Hodge structure ) is a finitely generated free module (or a finitely generated vector space) with a Hodge decomposition of (or ), so that the weight decomposition is defined by.
Examples
Hodge-Tate structures
Z (n)
is the whole Hodge structure with module
and . It is the only 1-dimensional Hodge structure weighing -2.
With becomes the -fold tensor product
designated.
Q (n)
is the rational Hodge structure with -Vector space
and .
is -fold tensor product
.
R (n)
is the Hodge structure with - vector space
and .
is -fold tensor product
.
Hodge decomposition theorem for Kähler manifolds
The cohomology of a compact Kähler manifold has a Hodge structure: according to Hodge's theorem one can identify the -th cohomology with the space of harmonic differential forms and it holds
where denotes the harmonic (p, q) forms . It applies .
Hodge filtration
To a pure Hodge structure by weight one describes the filtration
With
as an associated Hodge filtration.
The Hodge filtration determines the Hodge decomposition
The existence of a pure Hodge decomposition of weight is therefore equivalent to the existence of a filtration of with for sufficiently large and
for everyone with .
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